By simple modifications to the proof of Lemma \(3.9\)  of [1]}, the virtual dimension of \(\mathcal {M}(X,\tau )\)  is \(k-2\) . After enumerating the edges containing the unique \(k+1\)  valent vertex \(v_{out}\)  by \(e_0 = L_{out},e_1,\ldots ,e_k\) , by cutting the edges \(e_1,\ldots ,e_k\) , we produce a morphism \(\delta : \mathcal {M}(X,\tau ) \rightarrow \prod \mathcal {M}(X,\tau _i)\) , where for \(i>0\) , \(\tau _i\)  is the tropical type produced by cutting along edge \(e_i\)  which does not contain the vertex \(v_{out}\) , and \(\tau _0\)  is the resulting tropical type with a unique vertex \(v_{out}\)  with \(k+1\)  legs. It quickly follows from the definition of a \(k\) -trace type that \(\tau _i\)  is a decorated broken line type for \(i>0\) . The virtual dimensions of the moduli spaces \(\mathcal {M}(X,\tau _i)\)  are all 0 for \(i > 0\) , and \(k-2\)  for \(i = 0\) . By [2]} Theorem \(C\) , the cutting morphism \(\delta \)  is representable and finite. The following proposition gives a description of the class \(\delta _*([\mathcal {M}(X,\tau )]^{vir})\) . This closely follows [1]} Lemma \(6.6\) :
